Abstract
Motivated by the inconvenience or even inability to explain the mathematics of the state space optimization of finite state machines (FSMs) in most existing results, we consider the problem by viewing FSMs as logical dynamic systems. Borrowing ideas from the concept of equilibrium points of dynamic systems in control theory, the concepts of t-equivalent states and t-source equivalent states are introduced. Based on the state transition dynamic equations of FSMs proposed in recent years, several mathematical formulations of t-equivalent states and t-source equivalent states are proposed. These can be analogized to the necessary and sufficient conditions of equilibrium points of dynamic systems in control theory and thus give a mathematical explanation of the optimization problem. Using these mathematical formulations, two methods are designed to find all the t-equivalent states and t-source equivalent states of FSMs. Further, two ways of reducing the state space of FSMs are found. These can be implemented without computers but with only pen and paper in a mathematical manner. In addition, an open question is raised which can further improve these methods into unattended ones. Finally, the correctness and effectiveness of the proposed methods are verified by a practical language model.
摘要
现有大多数关于有限状态自动机 (finite state machines, FSM) 状态空间的优化方法不便甚至不能给出优化的数学意义. 本文将FSM视为逻辑动态系统, 借鉴控制论中动态系统平衡点的概念, 引入t-等价状态和t-源等价状态概念. 基于近年提出的FSM状态转移动力学方程, 得到t-等价状态和t-源等价状态的数学描述 (该数学描述可类比于控制论中关于动态系统平衡点的充要条件), 进而给出该优化问题的数学解释. 基于这些数学描述, 设计了求解FSM所有t-等价状态和t-源等价状态的两种方法. 此外, 找到降低FSM状态空间的两种路径. 可不借助计算机, 仅用纸笔以数学推演方式实现. 并且, 为使所设计的方法借助计算机能完全以无人值守方式运行, 提出一个开放性问题. 最后, 采用实际语言模型验证了结论的正确性和有效性.
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Acknowledgements
The authors would like to express their sincere thanks for the advice given by Assistant Prof. Xiangru XU and Prof. Yiguang HONG.
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Jumei YUE raised the research questions and the ideas to solve them. Yongyi YAN and Zengqiang CHEN guided Jumei YUE and He DENG to carry out the research in the whole process. Jumei YUE drafted the manuscript. He DENG helped organize the manuscript. Yongyi YAN and Zengqiang CHEN revised and finalized the paper.
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Jumei YUE, Yongyi YAN, Zengqiang CHEN, and He DENG declare that they have no conflict of interest.
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Project supported by the National Natural Science Foundation of China (Nos. U1804150, 62073124, and 61973175)
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Yue, J., Yan, Y., Chen, Z. et al. State space optimization of finite state machines from the viewpoint of control theory. Front Inform Technol Electron Eng 22, 1598–1609 (2021). https://doi.org/10.1631/FITEE.2000608
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DOI: https://doi.org/10.1631/FITEE.2000608
Key words
- Finite state machines
- Finite-valued systems
- Logical systems
- Logical networks
- Semi-tensor product of matrices
- Space optimization